A circle's center is at (4 ,2 ) and it passes through (6 ,7 ). What is the length of an arc covering (5pi ) /3 radians on the circle?

1 Answer
Tony B
Apr 6, 2016

Arc length~~28.2 to 1 decimal place

Explanation:

Let the radius of the circle be r
Let the length of arc be L_a

Distance from the circles centre to any point on its circumference is always the same.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Determine the radius of the circle")

=> r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

=> r = sqrt((6-4)^2 +(7-2)^2)

=>color(blue)(r = sqrt(29))" "-> 29 is a prime number so can not be simplified

'~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Determine the length of arc")

color(brown)("Important point:")

color(brown)("1 radian is such that its length of ark is the same as") color(brown)("the length of the radius.")

So the length of arc L_a=rxx (5pi)/3

=> L_a=sqrt(29)xx (5pi)/3

color(blue)(L_a~~28.2" to 1 decimal place")

'~~~~~~~~~~~~~~~~~~~~~~~~~

color(blue)("Check:")

Circumference = piD = pixx2sqrt(29)

and we have 1 2/3" of "1/2 of the circumference

=>L_a=1 2/3 xxpisqrt(29) =28.19... Confirmed!

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